In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest details, we "prove" wrong statements here and then. I plead guilty, having published myself one or two false (fortunately minor) papers.

So far, this is not harmful. The research community is able to point out incorrect statements, at least among those which have some importance in the development of mathematics. In time, the errors are fixed; this is the role of monographs to present a universally accepted state of the art of a topic.

But sometimes, hopefully rarely, the technicalities are such that a consensus does not emerge and a controversy raises, between the author and their critics. I have an example in the realm of wave stability in PDE models for fluid dynamics. The controversy has lasted for a decade or two and I don't see how it can be resolved some day; it could just kill the topic.

Are there famous endless controversies about the correctness of a significant paper? How did they influence their respective topics?

In this question, I am not concerned with other kinds of controversy, about priority or citations.

In principle, a mathematical paper should be complete and correct. New statements should be supported by appropriate proofs. But this is only theory. Because we often cannot enter into the smallest details, we "prove" wrong statements here and then. I plead guilty, having published myself one or two false (fortunately minor) papers.

So far, this is not harmful. The research community is able to point out incorrect statements, at least among those which have some importance in the development of mathematics. In time, the errors are fixed; this is the role of monographs to present a universally accepted state of the art of a topic.

But sometimes, hopefully rarely, the technicalities are such that a consensus does not emerge and a controversy raises, between the author and their critics. I have an example in the realm of wave stability in PDE models for fluid dynamics. The controversy has lasted for a decade or two and I don't see how it can be resolved some day; it could just kill the topic.

Are there famous endless controversies about the correctness of a significant paper? Are there some significant mathematical questions, that remain unsettled because people disagree on the status of released proofs? What should we do in order to salvage mathematical topics that suffer such tensions?

In this question, I am not concerned with other kinds of controversy, about priority or citations.